| L(s) = 1 | − 9.32·3-s + 1.51·5-s + 26.3·7-s + 59.9·9-s − 23.8·11-s − 35.7·13-s − 14.0·15-s − 116.·17-s + 31.6·19-s − 245.·21-s + 181.·23-s − 122.·25-s − 306.·27-s − 118.·29-s + 257.·31-s + 222.·33-s + 39.8·35-s + 330.·37-s + 333.·39-s − 126.·41-s − 197.·43-s + 90.5·45-s − 88.4·47-s + 351.·49-s + 1.08e3·51-s + 456.·53-s − 35.9·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 0.135·5-s + 1.42·7-s + 2.21·9-s − 0.653·11-s − 0.762·13-s − 0.242·15-s − 1.66·17-s + 0.381·19-s − 2.55·21-s + 1.64·23-s − 0.981·25-s − 2.18·27-s − 0.761·29-s + 1.49·31-s + 1.17·33-s + 0.192·35-s + 1.46·37-s + 1.36·39-s − 0.483·41-s − 0.701·43-s + 0.299·45-s − 0.274·47-s + 1.02·49-s + 2.97·51-s + 1.18·53-s − 0.0882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 9.32T + 27T^{2} \) |
| 5 | \( 1 - 1.51T + 125T^{2} \) |
| 7 | \( 1 - 26.3T + 343T^{2} \) |
| 11 | \( 1 + 23.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 456.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 497.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 122.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 915.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 145.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 199.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 807.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 254.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.332930602022784500178896038168, −8.137866710470162561177716558911, −7.29442074723461143114391137307, −6.53141611814979185011558672938, −5.50685018890363618728496152612, −4.89587371029545588659912190706, −4.37137358793220545300072376923, −2.35883607314296568192638813735, −1.18486395416296258048129854810, 0,
1.18486395416296258048129854810, 2.35883607314296568192638813735, 4.37137358793220545300072376923, 4.89587371029545588659912190706, 5.50685018890363618728496152612, 6.53141611814979185011558672938, 7.29442074723461143114391137307, 8.137866710470162561177716558911, 9.332930602022784500178896038168
